1
vote
0answers
12 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
2
votes
1answer
42 views

Stochastic integration by parts formula to prove identity between iterated integrals

if $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
1
vote
1answer
53 views

An exponential martingale [closed]

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
0
votes
1answer
30 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
1
vote
1answer
95 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
3
votes
1answer
29 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
0
votes
1answer
54 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
2
votes
1answer
42 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
1
vote
1answer
28 views

Stochastic Integrals and Martingales

I am attempting the following proof but two aspects of the solution confuse me: Given \begin{align} I^{n}_{t} = \int^t_0 \Delta_u^ndW_u = \sum_{j=0}^{k-1}\Delta_{t_{j}}(W_{t_{j+1}}-W_{t_{j}}) + ...
4
votes
0answers
92 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
0
votes
1answer
43 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
0
votes
1answer
14 views

Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
5
votes
1answer
90 views

Expectation of a stochastic integral

Let $M$ be a right-continuous local martingale, $s,t$ two times (stopping times, if you like). Under what conditions does the following hold: $$E\left(\int_s^t X \, dM\mid\mathcal{F}_s\right)\le 0$$ ...
0
votes
0answers
31 views

Defining the Radon-Nikodym as a solution to an SDE

Can someone please clarify this to me: If I have the Radon-Nikodym $L_t=\frac{dQ}{dP}$, on $\mathcal{F}_t$, then I know that $L_t$ is a non-negative P-martingale. So in many textbooks they say it is ...
0
votes
0answers
53 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
1
vote
0answers
49 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
0
votes
1answer
158 views

Preservation of Martingale property

Can someone help me to prove this? If possible I'd like the prove can avoid the use of local martingale. Prove the Ito integral $\int_0^T \Delta_t(\omega) dW_t(\omega)$ is a martingale if $E[\int_0^T ...
2
votes
1answer
137 views

Girsanov transformation and preservation of independence

If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original ...
1
vote
0answers
46 views

Supermartingale Lemma + related problems

Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
1
vote
1answer
106 views

Martingale inequality

Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, ...
0
votes
1answer
316 views

Show that this continuous local martingale is a martingale

We are given the following SDE: $$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$ and $$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$ We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
0
votes
0answers
76 views

Local martingale iff each component is a local martingale?

This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
3
votes
1answer
395 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
2
votes
2answers
180 views

Is the solution to a driftless SDE with Lipschitz variation a martingale?

If $\sigma$ is Lipschitz, with Lipschitz constant $K$, and $(X_t)_{t\geq 0}$ solves $$dX_t=\sigma(X_t)dB_t,$$ where $B$ is a Brownian motion, then is $X$ a martingale? I'm having difficulty getting ...
3
votes
2answers
113 views

If $X$ is a martingale, $X(0)=0$; $f$ left continuous, is $\int f X$ dt also a martingale?

If $X(t)$ is a martingale, and $X(0) = 0$. $f(t)$ is a left continuous function, $$ g(t) = \int_0^t f(s) X(s) ds $$ is $g(t)$ also a martingale? I guess it shall be, but don't know how to prove ...
0
votes
1answer
326 views

$d$-Dimensional Brownian Motion Martingales

Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a ...
4
votes
1answer
519 views

Martingale problem

If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent: (i) for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) ...
6
votes
1answer
729 views

Is this a martingale?

Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution $$ ...